# Subgroup and quotient

Algebra Level 5

Let $$G$$ be a group and $$N$$ a normal subgroup. Which of the following statements is/are always true?

I. If $$N$$ is finite and $$G/N$$ is finite, then $$G$$ is finite.

II. If $$N$$ is finite and cyclic and $$G/N$$ is finite and cyclic, then $$G$$ is finite and cyclic.

III. If $$N$$ is abelian and $$G/N$$ is abelian, then $$G$$ is abelian.

Notation:

A finite cyclic group is a group that is isomorphic to $${\mathbb Z}_n,$$ the integers mod $$n,$$ for some $$n.$$

An abelian group is a group whose operation is commutative: $$x * y = y * x$$ for all $$x,y \in G.$$

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